GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 10 Dec 2018, 00:43

# Dec 10th is GMAT Club's BDAY :-)

Free GMAT Club Tests & Quizzes for 24 hrs to celebrate together!

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

## Events & Promotions

###### Events & Promotions in December
PrevNext
SuMoTuWeThFrSa
2526272829301
2345678
9101112131415
16171819202122
23242526272829
303112345
Open Detailed Calendar
• ### Free lesson on number properties

December 10, 2018

December 10, 2018

10:00 PM PST

11:00 PM PST

Practice the one most important Quant section - Integer properties, and rapidly improve your skills.
• ### Free GMAT Prep Hour

December 11, 2018

December 11, 2018

09:00 PM EST

10:00 PM EST

Strategies and techniques for approaching featured GMAT topics. December 11 at 9 PM EST.

# M06-16

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 51055

### Show Tags

15 Sep 2014, 23:27
00:00

Difficulty:

85% (hard)

Question Stats:

47% (01:00) correct 53% (01:18) wrong based on 182 sessions

### HideShow timer Statistics

If $$\frac{t}{u} = \frac{x}{y}$$ and $$\frac{t}{y} = \frac{u}{x}$$ and $$t$$, $$u$$, $$x$$, and $$y$$ are non-zero integers, which of the following is true?

A. $$\frac{t}{u}=1$$
B. $$\frac{y}{x}=-1$$
C. $$t = -u$$
D. $$t = \pm u$$
E. None of the above

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 51055

### Show Tags

15 Sep 2014, 23:27
1
5
Official Solution:

If $$\frac{t}{u} = \frac{x}{y}$$ and $$\frac{t}{y} = \frac{u}{x}$$ and $$t$$, $$u$$, $$x$$, and $$y$$ are non-zero integers, which of the following is true?

A. $$\frac{t}{u}=1$$
B. $$\frac{y}{x}=-1$$
C. $$t = -u$$
D. $$t = \pm u$$
E. None of the above

Given that: $$\frac{t}{u} = \frac{x}{y}$$ and $$\frac{t}{y} = \frac{u}{x}$$.

So, $$\frac{t}{u} = \frac{x}{y}$$ and $$\frac{t}{u} = \frac{y}{x}$$ (from 2), which means that $$\frac{t}{u}$$ and $$\frac{x}{y}$$ equal to their reciprocals: $$\frac{t}{u}=\frac{u}{t}$$ and $$\frac{x}{y}=\frac{y}{x}$$. So, $$t^2=u^2$$ and $$t^2=u^2$$, which gives $$|t|=|u|$$ (or which is the same $$t = \pm u$$) and $$|x|=|y|$$ (or which is the same $$x = \pm y$$).

_________________
Manager
Joined: 03 Oct 2014
Posts: 132
Location: India
Concentration: Operations, Technology
GMAT 1: 720 Q48 V40
WE: Engineering (Aerospace and Defense)

### Show Tags

13 Feb 2015, 21:18
1
Eqn 1....Let t/u = x/y = k

Therefore t = uk, x = yk.....

Substitute in eqn 2...t/y = u/x

uk/y = u/yk

k^2 = 1

k = +/-1

so t = +/- u
Manager
Joined: 27 Jan 2013
Posts: 175
Location: United States
Concentration: Entrepreneurship, Marketing
GMAT 1: 730 Q49 V40
GPA: 3.5
WE: Supply Chain Management (Telecommunications)

### Show Tags

17 Feb 2015, 10:55
Bunuel In questions where they ask which is true.. are they asking for must be true or which can be true?

Asking since i solved this question as which can be true and stopped after choosing A as A was coming as true for a case where all numbers were equal.
Math Expert
Joined: 02 Sep 2009
Posts: 51055

### Show Tags

17 Feb 2015, 10:57
1
qw1981 wrote:
Bunuel In questions where they ask which is true.. are they asking for must be true or which can be true?

Asking since i solved this question as which can be true and stopped after choosing A as A was coming as true for a case where all numbers were equal.

Which is true = which must be true.
_________________
Intern
Joined: 05 Dec 2013
Posts: 29

### Show Tags

10 Jun 2015, 10:54
Why are we cross-multiplying ? The question does not state that integers t,x,u,v are positive...
Math Expert
Joined: 02 Sep 2009
Posts: 51055

### Show Tags

10 Jun 2015, 10:56
Randude wrote:
Why are we cross-multiplying ? The question does not state that integers t,x,u,v are positive...

It does not matter for equations, it does only for inequalities.
_________________
Intern
Joined: 18 Oct 2015
Posts: 2
GMAT 1: 720 Q47 V41
GPA: 3.98

### Show Tags

11 Dec 2015, 04:31
Hi,

I did the following:

From (1), ty=xu (cross multiplying)
From (2), tx=uy

so adding each side of the equalities, t(x+y)=u(x+y) so t=u (and not t= +-u)

What am I doing wrong?
Manager
Joined: 16 Feb 2016
Posts: 52
Concentration: Other, Other

### Show Tags

26 Apr 2016, 17:39
m9338 wrote:
Hi,

I did the following:

From (1), ty=xu (cross multiplying)
From (2), tx=uy

so adding each side of the equalities, t(x+y)=u(x+y) so t=u (and not t= +-u)

What am I doing wrong?

t=u
would be true for negative and positive values of u.
But I can see that some people would choose E.
Intern
Joined: 20 Jul 2012
Posts: 25

### Show Tags

15 Jun 2016, 01:34
2
m9338 wrote:
Hi,

I did the following:

From (1), ty=xu (cross multiplying)
From (2), tx=uy

so adding each side of the equalities, t(x+y)=u(x+y) so t=u (and not t= +-u)

What am I doing wrong?

One explanation that i can offer here is as follows -

The equation t(x+y)=u(x+y) holds good in almost all case, however (given x,y are non zero integers), consider a scenario where x= -y.
In this case you cannot say that t=u as you cant divide both sides with (x+y) !!! why? coz its 0

Incidentally this is one of the wrong answer choices as well
_________________

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Mega collection of All RCs from GMAT, GRE & LSAT https://gmatclub.com/forum/mega-rc-collection-from-gmat-gre-lsat-273512.html
Mega collection of CRs from various sources (including LSAT) https://gmatclub.com/forum/mega-critical-reasoning-cr-collection-274044.html

Intern
Joined: 21 Oct 2014
Posts: 4
Location: Ukraine
GMAT 1: 680 Q48 V35
GPA: 3.05

### Show Tags

18 Feb 2017, 06:53
I still can't understand why option D is correct, even though the question is MUST BE TRUE.
Here is my thinking:

Let's say t, u, x and y are all equal to 1 (non-zero integer). Then according to option D: 1 = +-1, which is only true for positive sign, but not negative. I mean the option D would be a MUST BE TRUE answer only if it was written like this:
t = u OR t = -u.
As far as I understand t = +-u means 't = u AND t = -u'
Intern
Joined: 04 Apr 2018
Posts: 1

### Show Tags

10 Jul 2018, 18:42
why is |t| = |u| the same thing as t = ±u? I don't understand that. Can someone please explain?
Math Expert
Joined: 02 Sep 2009
Posts: 51055

### Show Tags

10 Jul 2018, 19:27
gdhume wrote:
why is |t| = |u| the same thing as t = ±u? I don't understand that. Can someone please explain?

|t| = |u| means that the distance from t to 0 is the same as the distance from u from 0. So, either t = s, or t = -s.
_________________
Intern
Joined: 11 Feb 2013
Posts: 6
GMAT 1: 730 Q47 V42
GMAT 2: 740 Q49 V41

### Show Tags

13 Aug 2018, 06:24
m9338 wrote:
Hi,

I did the following:

From (1), ty=xu (cross multiplying)
From (2), tx=uy

so adding each side of the equalities, t(x+y)=u(x+y) so t=u (and not t= +-u)

What am I doing wrong?

I still cannot understand why this is wrong. Could someone help me?
Intern
Joined: 29 Jun 2017
Posts: 12

### Show Tags

05 Sep 2018, 05:40
So, t^2=u^2 and t^2=u^2 gives |t|=|u|

Could someone please elaborate the result derived above?
Math Expert
Joined: 02 Sep 2009
Posts: 51055

### Show Tags

05 Sep 2018, 05:45
Megha1119 wrote:
So, t^2=u^2 and t^2=u^2 gives |t|=|u|

Could someone please elaborate the result derived above?

By taking the square root.

Remember: $$\sqrt{x^2}=|x|$$.

The point here is that square root function can not give negative result: wich means that $$\sqrt{some \ expression}\geq{0}$$.

So $$\sqrt{x^2}\geq{0}$$. But what does $$\sqrt{x^2}$$ equal to?

Let's consider following examples:
If $$x=5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=x=positive$$;
If $$x=-5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=-x=positive$$.

So we got that:
$$\sqrt{x^2}=x$$, if $$x\geq{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<0$$.

What function does exactly the same thing? The absolute value function! That is why $$\sqrt{x^2}=|x|$$
_________________
Manager
Joined: 07 Aug 2018
Posts: 104
Location: United States (MA)
GMAT 1: 560 Q39 V28

### Show Tags

07 Nov 2018, 09:26
Hi Bunuel and chetan2u,

I am wondering if my approach is correct? Could you verify please?

$$\frac{T}{U}$$=$$\frac{X}{Y}$$ --> $$TY-XU=0$$

$$\frac{T}{Y}$$=$$\frac{U}{X}$$ --> $$TX-UY=0$$

Combining the 2 equations:

$$TY-XU=TX-UY$$
$$TY+UY=TX+XU$$
$$Y*(T+U)=X*(T+U)$$

Can I correctly infern that either $$|Y|=|X|$$ or $$(T+U)=0$$, therefore $$|T|=|U|$$?

_________________

Flashcards Quant + Verbal:https://gmatclub.com/forum/gmat-flashcards-108651.html
Thursdays with Ron:https://gmatclub.com/forum/manhattan-s-thursdays-with-ron-consolidated-video-index-223612.html#p2138753

Manager
Joined: 31 Mar 2018
Posts: 54

### Show Tags

07 Nov 2018, 09:37
m9338 wrote:
Hi,

I did the following:

From (1), ty=xu (cross multiplying)
From (2), tx=uy

so adding each side of the equalities, t(x+y)=u(x+y) so t=u (and not t= +-u)

What am I doing wrong?

Hi,

You will be right if (x+y) not equals to zero. But the stem says all are non zero integers that doesnt mean all are positive. so it is still possible that x is positive and y is negative with equal magnitude (or vice versa). Hence we can't just cancel out the x+y . Hope it helps.

Kudos=Thanks
Intern
Joined: 18 Jul 2018
Posts: 14

### Show Tags

01 Dec 2018, 03:49
Bunuel,

Hi..I have a doubt in this problem. t/u=x/y and t/u=y/x which basically means that t/u is equal to x/y as well as its reciprocal (y/x) which only means that x and y has to be of the same value making t/u =1 (Option A).

2ndly, y/x=-1 which also makes x/y = -1 therby t/u=x/y and also its reciprocal (y/x) thus option B also seems plausible.

I understood your explanation for Option D but Id like to understand the gap in my reasoning if I were to choose either A or B. Thankyou
Math Expert
Joined: 02 Sep 2009
Posts: 51055

### Show Tags

01 Dec 2018, 06:01
yashna36 wrote:
Bunuel,

Hi..I have a doubt in this problem. t/u=x/y and t/u=y/x which basically means that t/u is equal to x/y as well as its reciprocal (y/x) which only means that x and y has to be of the same value making t/u =1 (Option A).

2ndly, y/x=-1 which also makes x/y = -1 therby t/u=x/y and also its reciprocal (y/x) thus option B also seems plausible.

I understood your explanation for Option D but Id like to understand the gap in my reasoning if I were to choose either A or B. Thankyou

x/y = y/x does not mean that x = y. Cross-multiply: x^2 = y^2 --> |x| = |y|. Which means that x and y are same distance apart from 0: x = y or x = -y.
_________________
Re: M06-16 &nbs [#permalink] 01 Dec 2018, 06:01
Display posts from previous: Sort by

# M06-16

Moderators: chetan2u, Bunuel

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.